Error Estimates of the Crank-Nicolson-Polylinear FEM with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped
We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded n–dimensional parallelepiped (\(n\ge 1\)). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transparent boundary conditions is considered. We present its stability properties and derive new error estimates \(O(\tau ^2+|h|^2)\) uniformly in time in \(L^2\) space norm, for \(n\ge 1\), and mesh \(H^1\) space norm, for \(1\le n\le 3\) (a superconvergence result), under the Sobolev-type assumptions on the initial function. Such estimates are proved for methods with the discrete TBCs for the first time.
KeywordsTime-dependent Schrödinger equation Unbounded domain Crank-Nicolson scheme Finite element method Discrete transparent boundary conditions Stability Error estimates Superconvergence
The study is supported by The National Research University – Higher School of Economics’ Academic Fund Program in 2014–2015, research grant No. 14-01-0014.
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